Exponent laws compress repeated multiplication. Each rule has a specific structure: some combine powers sharing a base, while others distribute a power across an entire product or quotient. Scientific notation uses those same rules to represent very large and very small values efficiently.
Learning objectives
- Distinguish product, quotient, and nested-power structures before applying a law.
- Rewrite zero and negative exponents while respecting nonzero-base conditions.
- Convert between decimal and scientific notation.
- Multiply and divide scientific-notation values and normalize the coefficient.
- Detect invalid exponent operations across addition or subtraction.
The exponent laws
| Law | Formula | Conditions | Example | Common mistake |
|---|---|---|---|---|
| Product of powers | \(a^m a^n=a^{m+n}\) | Same base | \(y^4y^3=y^7\) | Multiplying the exponents |
| Power of a power | \((a^m)^n=a^{mn}\) | Nested powers | \((z^2)^5=z^{10}\) | Adding the exponents |
| Power of a product | \((ab)^m=a^mb^m\) | Power applies to the whole product | \((2xy^2)^3=8x^3y^6\) | Applying the power to only one factor |
| Quotient of powers | \(\frac{a^m}{a^n}=a^{m-n}\) | \(a\ne0\) | \(\frac{p^9}{p^4}=p^5\) | Subtracting in the wrong order |
| Power of a quotient | \((\frac{a}{b})^m=\frac{a^m}{b^m}\) | \(b\ne0\) | \((\frac{3x}{4})^2=\frac{9x^2}{16}\) | Leaving the denominator unpowered |
| Zero exponent | \(a^0=1\) | \(a\ne0\) | \((-7q)^0=1\) | Writing zero |
| Negative exponent | \(a^{-n}=\frac{1}{a^n}\) | \(a\ne0\) | \(5^{-2}=\frac{1}{25}\) | Treating the result as negative |
Why the product law works
Combine repeated factors
Simplify \(4x^3y^2\cdot(-2x^5y)\).
- Coefficients
Multiply \(4(-2)=-8\).
- Shared bases
Combine \(x^3x^5=x^8\) and \(y^2y=y^3\).
Nested and distributed powers
Power every factor
Simplify \((-3a^2b^3)^2\).
- Distribute the outside power
Square the coefficient and each variable power.
- Multiply nested exponents
\((-3)^2=9\), \((a^2)^2=a^4\), and \((b^3)^2=b^6\).
Quotients, zero exponents, and negative exponents
Simplify a negative power of a fraction
Simplify \((\frac{-2x}{5y})^{-3}\), where \(x\ne0\) and \(y\ne0\).
- Take the reciprocal
Rewrite as \((\frac{5y}{-2x})^3\).
- Apply the power
Cube the numerator and denominator, preserving the negative sign.
Scientific notation
- Scientific notation
- A representation \(a\times10^n\), where \(n\) is an integer and the coefficient satisfies \(1\le |a|<10\).
Track the decimal shift
The decimal moves 5 places right; the exponent records the direction and distance.
- Original number0.000064
- Move the decimal5 places right
- Normalized coefficient6.4
- Exponent\(-5\)
\[6.4\times10^{-5}\]
| Starting form | Decimal movement | Exponent sign | Example |
|---|---|---|---|
| Large decimal number | Move left | Positive | \(438000=4.38\times10^5\) |
| Small nonzero decimal | Move right | Negative | \(0.000091=9.1\times10^{-5}\) |
| Positive power of ten | Move right to expand | Positive | \(7.2\times10^4=72000\) |
| Negative power of ten | Move left to expand | Negative | \(3.6\times10^{-3}=0.0036\) |
Convert a decimal to scientific notation
- Normalize
Move the decimal so the coefficient has absolute value from \(1\) up to but not including \(10\).
- Count
Count how many places the decimal moved.
- Choose the sign
Moving left gives a positive exponent; moving right gives a negative exponent.
- Check
Expand \(10^n\) mentally to confirm the original scale.
Operations in scientific notation
Multiply, then normalize
Simplify \((6\times10^4)(3.5\times10^{-2})\) in scientific notation.
- Coefficients
Multiply \(6(3.5)=21\).
- Powers of ten
Add exponents: \(10^4\cdot10^{-2}=10^2\).
- Normalize
\(21\times10^2=2.1\times10^3\).
Divide coefficients and subtract exponents
Simplify \(\frac{8.4\times10^7}{2.1\times10^3}\).
- Coefficients
\(8.4\div2.1=4\).
- Powers of ten
\(10^7\div10^3=10^{7-3}=10^4\).
Common mistakes and traps
Quick error audit
Structure
Confirm whether powers are multiplied, divided, or nested before changing exponents.
Reciprocal
A negative exponent moves the powered factor across the fraction bar.
Scale
Large values normally have positive scientific exponents; small nonzero decimals normally have negative exponents.
Normalization
Finish with \(1\le|a|<10\), not \(15\times10^7\) or \(0.4\times10^6\).
Normalize the result
Write \((4\times10^6)(7\times10^{-3})\) in normalized scientific notation.
- \(28\times10^3\)
- \(2.8\times10^4\)
- \(2.8\times10^3\)
- \(28\times10^9\)
Show answer and explanation
Answer: \(2.8\times10^4\)
Multiply coefficients to get \(28\), add exponents to get \(10^3\), then normalize \(28\times10^3=2.8\times10^4\).
What to remember
- Product adds exponents, quotient subtracts, and power of a power multiplies.
- A power distributes across every factor in a product or quotient.
- Zero and negative exponent rules require nonzero bases.
- Negative exponent means reciprocal, not negative value.
- Scientific notation is complete only after its coefficient is normalized.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.